Recently, the information-theoretical framework has been proven to be able to obtain non-vacuous generalization bounds for large models trained by Stochastic Gradient Langevin Dynamics (SGLD) with isotropic noise. In this paper, we optimize the information-theoretical generalization bound by manipulating the noise structure in SGLD. We prove that with constraint to guarantee low empirical risk, the optimal noise covariance is the square root of the expected gradient covariance if both the prior and the posterior are jointly optimized. This validates that the optimal noise is quite close to the empirical gradient covariance. Technically, we develop a new information-theoretical bound that enables such an optimization analysis. We then apply matrix analysis to derive the form of optimal noise covariance. Presented constraint and results are validated by the empirical observations.

We present a variational method for online state estimation and parameter learning in state-space models (SSMs), a ubiquitous class of latent variable models for sequential data. As per standard batch variational techniques, we use stochastic gradients to simultaneously optimize a lower bound on the log evidence with respect to both model parameters and a variational approximation of the states' posterior distribution. However, unlike existing approaches, our method is able to operate in an entirely online manner, such that historic observations do not require revisitation after being incorporated and the cost of updates at each time step remains constant, despite the growing dimensionality of the joint posterior distribution of the states. This is achieved by utilizing backward decompositions of this joint posterior distribution and of its variational approximation, combined with Bellman-type recursions for the evidence lower bound and its gradients. We demonstrate the performance of this methodology across several examples, including high-dimensional SSMs and sequential Variational Auto-Encoders.

We present a framework for transfer learning based on modular variational Gaussian processes (GP). We develop a module-based method that having a dictionary of well fitted GPs, one could build ensemble GP models without revisiting any data. Each model is characterised by its hyperparameters, pseudo-inputs and their corresponding posterior densities. Our method avoids undesired data centralisation, reduces rising computational costs and allows the transfer of learned uncertainty metrics after training. We exploit the augmentation of high-dimensional integral operators based on the Kullback-Leibler divergence between stochastic processes to introduce an efficient lower bound under all the sparse variational GPs, with different complexity and even likelihood distribution. The method is also valid for multi-output GPs, learning correlations a posteriori between independent modules. Extensive results illustrate the usability of our framework in large-scale and multi-task experiments, also compared with the exact inference methods in the literature.

The need for algorithms able to solve Reinforcement Learning (RL) problems with few trials has motivated the advent of model-based RL methods. The reported performance of model-based algorithms has dramatically increased within recent years. However, it is not clear how much of the recent progress is due to improved algorithms or due to improved models. While different modeling options are available to choose from when applying a model-based approach, the distinguishing traits and particular strengths of different models are not clear. The main contribution of this work lies precisely in assessing the model influence on the performance of RL algorithms. A set of commonly adopted models is established for the purpose of model comparison. These include Neural Networks (NNs), ensembles of NNs, two different approximations of Bayesian NNs (BNNs), that is, the Concrete Dropout NN and the Anchored Ensembling, and Gaussian Processes (GPs). The model comparison is evaluated on a suite of continuous control benchmarking tasks. Our results reveal that significant differences in model performance do exist. The Concrete Dropout NN reports persistently superior performance. We summarize these differences for the benefit of the modeler and suggest that the model choice is tailored to the standards required by each specific application.

Hierarchical forecasting problems arise when time series compose a group structure that naturally defines aggregation and disaggregation coherence constraints for the predictions. In this work, we explore a new forecast representation, the Poisson Mixture Mesh (PMM), that can produce probabilistic, coherent predictions; it is compatible with the neural forecasting innovations, and defines simple aggregation and disaggregation rules capable of accommodating hierarchical structures, unknown during its optimization. We performed an empirical evaluation to compare the PMM \ to other hierarchical forecasting methods on Australian domestic tourism data, where we obtain a 20 percent relative improvement.

The time-domain simulation is the fundamental tool for power system transient stability analysis. Accurate and reliable simulations rely on accurate dynamic component modeling. In practical power systems, dynamic component modeling has long faced the challenges of model determination and model calibration, especially with the rapid development of renewable generation and power electronics. In this paper, based on the general framework of neural ordinary differential equations (ODEs), a modified neural ODE module and a neural differential-algebraic equations (DAEs) module for power system dynamic component modeling are proposed. The modules adopt an autoencoder to raise the dimension of state variables, model the dynamics of components with artificial neural networks (ANNs), and keep the numerical integration structure. In the neural DAE module, an additional ANN is used to calculate injection currents. The neural models can be easily integrated into time-domain simulations. With datasets consisting of sampled curves of input variables and output variables, the proposed modules can be used to fulfill the tasks of parameter inference, physics-data-integrated modeling, black-box modeling, etc., and can be easily integrated into power system dynamic simulations. Some simple numerical tests are carried out in the IEEE-39 system and prove the validity and potentiality of the proposed modules.

The Stochastic Shortest Path (SSP) model refers to a type of reinforcement learning (RL) problems where an agent repeatedly interacts with a stochastic environment and aims to reach some specific goal state while minimizing the cumulative cost. Compared with other popular RL settings such as episodic and infinite-horizon Markov Decision Processes (MDPs), the horizon length in SSP is random, varies across different policies, and can potentially be infinite because the interaction only stops when arriving at the goal state. Therefore, the SSP model includes both episodic and infinitehorizon MDPs as special cases, and is comparably more general and of broader applicability. In particular, many goal-oriented real-world problems fit better into the SSP model, such as navigation and GO game (Andrychowicz et al., 2017; Nasiriany et al., 2019). In recent years, there emerges a line of works on developing efficient algorithms and the corresponding analyses for learning SSP. Most of them consider the episodic setting, where the interaction between the agent and the environment proceeds in K episodes (Cohen et al., 2020; Tarbouriech et al., 2020a). For tabular SSP models where the sizes of the action and state space are finite, Cohen et al. (2021) developed a finite-horizon reduction algorithm that achieves the minimax

Many of us (myself included) have felt discouraged from using Bayesian statistics for analysis. Supposedly, Bayesian statistics has a bad reputation: it is difficult and heavily dependent on math. Also, because of its relevance to many fields, Data Science included, writers and professionals, want to get a head start by publishing articles on how the formula works. I believe data professionals, academics, existing books, and online courses are responsible for creating the negative stereotype of Bayes' hard work. We can all agree that not everyone is attracted to mathematical formulas.

Box-Cox transformation is commonly used remedy when the normality is not met. This comherensive guide includes estimation techniques and use of Box-Cox transformation in practice. Find out how to apply Box-Cox transformation in R. In this tutorial, we will work on Box-Cox transformation in R. Firstly, we will mention two types of estimation techniques for Box-Cox transformation parameter. These are maximum likelihood estimation (MLE) and estimation via normality tests. Secondly, we will work how to apply Box-Cox transformation in practice.

It is desirable to combine the expressive power of deep learning with Gaussian Process (GP) in one expressive Bayesian learning model. Deep kernel learning showed success in adopting a deep network for feature extraction followed by a GP used as function model. Recently,it was suggested that, albeit training with marginal likelihood, the deterministic nature of feature extractor might lead to overfitting while the replacement with a Bayesian network seemed to cure it. Here, we propose the conditional Deep Gaussian Process (DGP) in which the intermediate GPs in hierarchical composition are supported by the hyperdata and the exposed GP remains zero mean. Motivated by the inducing points in sparse GP, the hyperdata also play the role of function supports, but are hyperparameters rather than random variables. We follow our previous moment matching approach to approximate the marginal prior for conditional DGP with a GP carrying an effective kernel. Thus, as in empirical Bayes, the hyperdata are learned by optimizing the approximate marginal likelihood which implicitly depends on the hyperdata via the kernel. We shall show the equivalence with the deep kernel learning in the limit of dense hyperdata in latent space. However, the conditional DGP and the corresponding approximate inference enjoy the benefit of being more Bayesian than deep kernel learning. Preliminary extrapolation results demonstrate expressive power from the depth of hierarchy by exploiting the exact covariance and hyperdata learning, in comparison with GP kernel composition, DGP variational inference and deep kernel learning. We also address the non-Gaussian aspect of our model as well as way of upgrading to a full Bayes inference.